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    <a class="article-category-link" href="/categories/高等数学/">高等数学</a>►<a class="article-category-link" href="/categories/高等数学/极限与连续/">极限与连续</a>
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      函数的极限
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        <h2 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h2><h3 id="自变量趋向于有限值"><a href="#自变量趋向于有限值" class="headerlink" title="自变量趋向于有限值"></a>自变量趋向于有限值</h3><h3 id="定义-1"><a href="#定义-1" class="headerlink" title="定义"></a>定义</h3><p>若y = f(x) 在x =a 的去心邻域内有定义, (总能找到$\delta$)</p>
<script type="math/tex; mode=display">
当 \forall\large\varepsilon > 0, \exists\delta > 0, 当 0 <|x-a| < \delta时</script><script type="math/tex; mode=display">
|f(x) - A| < \varepsilon</script><p>则称</p>
<script type="math/tex; mode=display">
\lim_{x\rightarrow a}f(x) = A</script><h3 id="要点"><a href="#要点" class="headerlink" title="要点"></a>要点</h3><script type="math/tex; mode=display">
 x \to a  \implies  x  \neq  a</script><script type="math/tex; mode=display">
\lim_{x\rightarrow a} f(x) 与 f(a) 无关</script><script type="math/tex; mode=display">
x \to a \implies \begin{cases} x \to a^- \\\\\ x \to a^+  \end{cases}</script><script type="math/tex; mode=display">
a的去心邻域, 邻域半径 \delta >  0,   \forall\varepsilon > 0, \exists\delta > 0, 当x \in (a - \delta, a) 时,</script><script type="math/tex; mode=display">
充要条件 \lim_{x\rightarrow a}f(x) \exists \Longleftrightarrow  f(a^-) , f(a^+) \exists 且相等</script><h3 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h3><h4 id="例1"><a href="#例1" class="headerlink" title="例1"></a>例1</h4><script type="math/tex; mode=display">
\lim_{x\rightarrow 2}(3x+1) = 7</script><p>证:  </p>
<script type="math/tex; mode=display">
对任意\varepsilon  > 0 \\\\  
若原式成立, 则需满足|f(x) - 7| = |3x+1-7| = |3x-6| = 3|x-2| < \large\varepsilon成立\\\\  
即满足 |x-2| <\frac{ \varepsilon }{3} 成立 \\\\  
取0 < \delta <= \frac{\varepsilon}{3} ,此时满足0 < |x - 2| < \delta <= \varepsilon\\\\  
故等式成立</script><h2 id="自变量趋向于无穷大时函数的极限"><a href="#自变量趋向于无穷大时函数的极限" class="headerlink" title="自变量趋向于无穷大时函数的极限"></a>自变量趋向于无穷大时函数的极限</h2><h3 id="情况-1"><a href="#情况-1" class="headerlink" title="情况 1"></a>情况 1</h3><p>区分正负无穷极限</p>
<script type="math/tex; mode=display">
y = arctanx \\\\  
\lim_{x\rightarrow -\infty} arctanx = -\frac{\pi}{2}</script><script type="math/tex; mode=display">
\lim_{x\rightarrow +\infty} arctanx = \frac{\pi}{2}</script><p><img src="../assets/1.svg" alt=""></p>
<h3 id="情况-2"><a href="#情况-2" class="headerlink" title="情况 2"></a>情况 2</h3><script type="math/tex; mode=display">
y = 2 + e^{-x^2}</script><p><img src="../assets/2.svg" alt="xc"></p>
<h3 id="定义-2"><a href="#定义-2" class="headerlink" title="定义"></a>定义</h3><script type="math/tex; mode=display">
if \quad\forall\varepsilon > 0, \exists X > 0, 当 |x| > X 时  \\\\  
|f(x) - A | < \lim_{x\rightarrow\infty} = \varepsilon,  \\\\</script><script type="math/tex; mode=display">
则 \lim_{x\rightarrow\infty} = A</script><h3 id="例-1"><a href="#例-1" class="headerlink" title="例 1"></a>例 1</h3><script type="math/tex; mode=display">
\lim_{x\rightarrow +\infty} \frac{1}{2x+1} = 0</script><script type="math/tex; mode=display">
证: 对于\forall\varepsilon > 0,若要求 
|f(x) - A | = |\frac{1}{2x+1} - 0| \\\\  
=|\frac{1}{2x+1}| < \varepsilon \\\\  
即 x > \frac{\frac{1}{\varepsilon} - 1}{2} \\\\  
取 X <= \frac{\frac{1}{\varepsilon} - 1}{2} 则满足 \\\\  
x > X, |f(x) - A | < \varepsilon, 从而得证</script>
      
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              <ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#定义"><span class="toc-number">1.</span> <span class="toc-text">定义</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#自变量趋向于有限值"><span class="toc-number">1.1.</span> <span class="toc-text">自变量趋向于有限值</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#定义-1"><span class="toc-number">1.2.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#要点"><span class="toc-number">1.3.</span> <span class="toc-text">要点</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例题"><span class="toc-number">1.4.</span> <span class="toc-text">例题</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#例1"><span class="toc-number">1.4.1.</span> <span class="toc-text">例1</span></a></li></ol></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#自变量趋向于无穷大时函数的极限"><span class="toc-number">2.</span> <span class="toc-text">自变量趋向于无穷大时函数的极限</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#情况-1"><span class="toc-number">2.1.</span> <span class="toc-text">情况 1</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#情况-2"><span class="toc-number">2.2.</span> <span class="toc-text">情况 2</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#定义-2"><span class="toc-number">2.3.</span> <span class="toc-text">定义</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#例-1"><span class="toc-number">2.4.</span> <span class="toc-text">例 1</span></a></li></ol></li></ol>
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